Random submanifolds and their Zonoid section

Abstract: I will present a new formalism to encode random hypersurfaces (or submanifolds) of a Riemannian manifold in terms of a continuous family of convex bodies in the cotangent bundle (or its exterior power). This family depends only on pointwise data and determines the average volume density and the average current of the random submanifold. Moreover, such convex bodies belong to a particular class, called Zonoids, which have recently been shown to have a multiplicatication structure. We will see that such structure corresponds to the intersection of random submanifolds, in a fashion similar to the cohomology ring.

Random geometry is a subject of increasing interest, due in particular to the use of probabilistic methods to study nodal sets of eigenfunctions of the Laplacian via Riemannian random waves. I will also present a brief survey of some facts in random geometry to convey the general idea that random structures can be used to describe classical deterministic objects in a new way, offering an alternative source of intuition. In particular, I will discuss how to generate unlimited versions of the Gauss-Bonnet-Chern theorem. This is based on a joint work with Léo Mathis, to appear.